1A Brief History of Sine, Cosine & Tangent
The trig ratios you use today were born from astronomy — and the path from ancient India to your calculator app is a 1,500-year international relay race.
The Indian mathematician and astronomer Aryabhata described what he called the jya (Sanskrit for "chord-half") in his astronomical text Aryabhatiya. His jya of an angle is essentially the modern sine — but expressed as a length on a circle of fixed radius, not yet as a pure ratio. He compiled a table of 24 jya values at 3¾° intervals, accurate enough for predicting planetary positions. This is the earliest known systematic table of what we now call sines.
Arab astronomers translated Aryabhata's jya as jiba, then abbreviated it. When later scribes wrote only the consonants (Arabic doesn't write short vowels), it became jb. European translators mistakenly read jb as the Arabic word jaib — meaning "bay" or "fold of a garment" — and translated it into Latin as sinus (a curve or bay). That is why we say sine.
Meanwhile, Al-Battani (~858–929) introduced the cosine (as the sine of the complementary angle) and introduced the tangent ratio, though he called it the "shadow" (because it corresponds to the shadow cast by a vertical gnomon).
Persian mathematician Abu'l-Wafa al-Buzjani introduced the tangent function explicitly and computed the first comprehensive table of tangents, sines, and cosines. He is also credited with deriving the modern form of the sine rule and introducing the secant and cosecant. His trigonometry was explicitly for a unit circle — making the values pure ratios, exactly as we use them today.
German astronomer Johannes Müller (known as Regiomontanus) published De triangulis omnimodis ("On Triangles of All Kinds") — the first book in Europe to treat trigonometry as an independent subject, not just a tool for astronomy. He presented the law of sines and systematized the use of all six trig functions. A generation later, Georg Rheticus (1514–1576) was the first to define trig ratios purely in terms of right-triangle side ratios (not arcs of circles), making SohCahToa possible.
Swiss mathematician Leonhard Euler standardized the notation
sin, cos, tan and —
crucially — redefined them as functions of real numbers rather
than just ratios inside a specific triangle. His formulation in
Introductio in Analysin Infinitorum unified trigonometry with
algebra and calculus. The unit circle, the identity
sin²(θ) + cos²(θ) = 1, and the endless decimal expansions
all flow from his framework. Modern scientific notation is essentially
Euler's notation.
The mnemonic SOH–CAH–TOA is a 20th-century classroom invention. It does not appear in any classical text; it was created by math teachers as a memory aid for the three ratios. Nobody is certain who coined it first — it surfaced in American math education around the mid-1900s and spread widely through textbooks and, later, the internet. Despite its humble origins it has become so universal that the acronym is recognized world-wide.
TIP: Some students remember it as "Some Old Hippo Caught A Hippo Trampling On Alligators." Make up your own!
2The Right Triangle — Sides and Angles
Trigonometric ratios are defined for acute angles in a right triangle. The names of the sides depend on which angle you are looking from.
Viewed from ∠A (lower-left): side a is opposite, side b is adjacent, side c is the hypotenuse.
| Side | In this app | Relative to ∠A | Meaning |
|---|---|---|---|
| a | BC — vertical leg | Opposite (opp) | The leg across from ∠A; does not touch vertex A. |
| b | CA — horizontal leg | Adjacent (adj) | The leg that forms ∠A (not the hypotenuse). |
| c | AB — hypotenuse | Hypotenuse (hyp) | Always opposite the right angle; always the longest side. |
3SohCahToa — The Three Ratios
Each syllable of SOH–CAH–TOA is a mini-formula:
Using the Ratios to Find a Missing Side
If you know one acute angle θ and one side, you can always find another side by choosing the ratio that connects the two sides you care about, then solving for the unknown.
Worked Examples — Side-Finding
| Given | Find | Which ratio? | Calculation | Answer |
|---|---|---|---|---|
| ∠A = 30°, c = 10 | a (opp) | SOH: sin(A) = a/c | a = 10 × sin(30°) = 10 × 0.5 | 5.00 |
| ∠A = 45°, c = 7 | b (adj) | CAH: cos(A) = b/c | b = 7 × cos(45°) = 7 × 0.7071… | 4.95 |
| ∠A = 53°, b = 6 | a (opp) | TOA: tan(A) = a/b | a = 6 × tan(53°) = 6 × 1.3270… | 7.96 |
| ∠A = 60°, a = 9 | c (hyp) | SOH: sin(A) = a/c → c = a/sin(A) | c = 9 / sin(60°) = 9 / 0.8660… | 10.39 |
opp & hyp → sin adj & hyp → cos opp & adj → tan
Finding ∠B After Finding ∠A
In any right triangle, the two acute angles are complementary — they add up to 90°:
So once you know ∠A, you get ∠B for free — no trig needed!
4Inverse Trig — Finding the Angle
The ratios SOH, CAH, TOA go from angle → side. The inverse functions go from side → angle:
arcsin (sin−1)
Use when you know opposite and hypotenuse.
arccos (cos−1)
Use when you know adjacent and hypotenuse.
arctan (tan−1)
Use when you know opposite and adjacent.
Worked Examples — Angle-Finding
| Given sides | Find ∠A | Which inverse? | Calculation | ∠A |
|---|---|---|---|---|
| a = 3, c = 5 | ∠A (opp/hyp) | arcsin | sin−1(3/5) = sin−1(0.6) | 36.87° |
| b = 12, c = 13 | ∠A (adj/hyp) | arccos | cos−1(12/13) = cos−1(0.923…) | 22.62° |
| a = 8, b = 15 | ∠A (opp/adj) | arctan | tan−1(8/15) = tan−1(0.5333…) | 28.07° |
| a = 7, c = 25 | ∠A (opp/hyp) | arcsin | sin−1(7/25) = sin−1(0.28) | 16.26° |
5Why Does It Matter?
Trig is not just a math-class topic — it is the hidden engine behind almost everything in the physical world that involves angles, waves, or rotation.
🏗 Architecture & Construction
Roof pitches, ramp grades, load angles on beams, and staircase rise-over-run all reduce to right-triangle problems. A carpenter who knows tangent can set any bevel angle without a protractor: run a known rise over a known run and use arctan.
📱 Computer Graphics & Game Engines
Every 2-D and 3-D transformation (rotation,
projection, lighting) in a game or app uses sin and cos. Moving a
sprite "forward" in the direction it faces requires
x += speed * cos(angle) and
y += speed * sin(angle). The entire GPU pipeline
is built on trig.
🌎 Navigation & GPS
Finding the shortest path between two points on the globe (the great-circle route) uses the haversine formula, which is built from arcsin and trigonometric identities. GPS receivers solve trig problems in real time to compute your latitude and longitude from satellite distances.
🎵 Sound & Waves
Sound, light, radio, and any other wave is described by sine functions: y = A sin(2πft + φ). Audio engineers, physicists, and EE students use trig daily. Fourier analysis — the math behind MP3 compression, phone calls, and MRI machines — is essentially "decompose any signal into sine waves."
🔭 Astronomy & Surveying
Trig was born in astronomy and still drives it. The distance to the Moon, the diameter of the Sun, the height of a mountain, and the distance across an uncrossable river can all be computed from angle measurements and a single known baseline length — no ruler required.
🤖 Robotics & Engineering
Robotic arms, CNC machines, and servo-driven mechanisms are programmed with inverse kinematics — given a desired endpoint position, find the joint angles. That is arctan at its core. Even the humble servo in your RC car uses trig to translate a pulse width into an angular position.
6Trig Explorer
Use the two modes below to practice the same types of problems you'll encounter in the Solve Triangles app.
Quick presets:
Quick presets (Pythagorean triples):